It’s wonderful to be here. I spend most of my days with people who don’t fully get me. Wife, friends, dog – none of them gets me like you get me.

None of them understands the feeling of mathematical epiphany that motivates my professional life, the sudden transition from not knowing to knowing.

One of my earliest mathematical epiphanies was the realization that if you let the number of sides on a regular polygon increase without bound, you get a circle.

And that all the relationships you find in a regular polygon have analogous relationships in a circle. For me, that realization was literally a religious experience. I finished that limit on the back of a church bulletin while a churchlady glared at me.

So on the one hand it’s great to be in this room – I am among my people – but on the other hand it’s really uncomfortable to be here because you all make me really aware of my privilege, and aware of how many people are not in this room.

The economic 1% gets a lot of grief lately and whether we know it or not, whether we like it or not, we are all also in the 1% – the mathematical 1%.

In 2014, 2.8 million degrees were awarded in US universities – bachelors, masters, and doctorates – and 1.1% of them were in mathematics. If you change the denominator to reflect not advanced degree holders but anyone with a high school diploma our elitism becomes even more apparent.

I was on Instagram last night checking out the #MAAthfest hashtag along with The Rich Kids of Instagram. While there are fewer yachts, bottles, and shrink wrapped stacks of bills on the left, and maybe more plaid and elbow patches, there is still the same exuberant sense of having arrived. We have made it.

And just as the economic 1% creates systems that preserve its status – policies like the mortgage interest deduction for homeowners, discriminatory lending policies, and lower taxes on capital gains than income – through our action or inaction we create systems that preserve our status as the knowers and doers of mathematics.

When someone says, “I’m not a math person,” what do you say back? Barring certain disabilities or exceptionalities, everyone starts life a math person. Infants can recognize changing quantity. Brazilian street vendors develop sophisticated arithmetic algorithms before they set foot in school.

It is our action and inaction that teach people they are not mathematical. So please consider taking two actions to extend your privilege to the other 99% of humanity.

First, change the definition of mathematics that people experience.

[Here we explored together Circle-Square, a task that involves questioning, estimation, intentionally declaring wrong answers, recalling what you know about circles and squares, computing an answer, and verifying it. You can watch it.]

Now I don’t want to suggest to you that this is the experience that will change a person’s definition of mathematics and extend our privilege to the 99%. I just want to suggest to you that you just had a very different mathematical experience than the people who encountered that problem in its original form:

Mark an arbitrary point P on a line segment AB. Let AP form the perimeter of a square and BP form the circumference of a circle. Find P such that the area of the square and circle are maximized.

That experience offers people only a certain kind of mathematical work. You recall what you know about perimeter, circumference, and area, compute it, and verify it in the back of the book.

Those verbs are our mortgage interest rate deduction, our discriminatory lending policy, and our tax advantages. Through our action and inaction, society has come to understand that math is a merry-go-round revolving endlessly through those three verbs – remember a procedure, compute it, verify it.

You might think, “Well that’s what math is,” but the definition of math isn’t a physical constant in the universe. It’s defined by people, just as people define the ways that wealth and power accrue in the world. That definition is then underlined, reflected, and enforced in public policy, curriculum, and syllabi.

So, second, let’s change the definition of mathematics in public policy, curriculum, and syllabi.

To begin with, let’s eliminate policies that require intermediate algebra for college study.

The facts as I understand them are that:

College completion is increasingly essential to even partial economic participation.

College study is generally predicated on a student’s ability to pass a mathematics entrance exam. In the California State University system, that exam is heavily weighted towards intermediate algebra, problems like these, the majority of which depend on the recollection of an obscure and abstract procedure:

Students fail these exams in staggering numbers (68% nationally) placing into “developmental math” courses, courses which cost time and money and don’t offer credit towards graduation.

Those courses are disproportionally composed of African American and Latinx students.

Only 32% of students in developmental math ever take a math course required for graduation.

It’s hard to imagine a machine more perfectly configured for the preservation of mathematical privilege.

Those statistics would bother me less if either a) I believed in the value of intermediate algebra, b) better alternatives weren’t available. Neither is true. That intermediate algebra has little value to the majority of college educated professionals hardly requires a defense. As Uri Treisman said, “The most common use of algebra in the adult world is helping their kids with algebra.”

I am sympathetic to the argument, however, that we shouldn’t choose college requirements solely because they’re useful professionally. College should offer students a broad survey of every discipline – a general education, as it’s called. That survey should generate intellectual interest where perhaps there was none; it should awaken students to intellectual possibilities they hadn’t considered; it should increase the likelihood they’ll speak favorably about the discipline after college.

Those goals are served poorly by intermediate algebra. And better alternatives to intermediate algebra exist to serve the CSU’s desire to “assess mathematical skills needed in CSU General Education (GE) programs in quantitative reasoning.”

Specifically, statistics.

When 907 CUNY students were assigned either to remedial algebra, remedial algebra and supplementary workshops, or college-level statistics and workshops, that latter group a) passed their course in greater numbers (earning credit!) and also b) accumulated more credits in later courses.

So we should be excited to see the California State University drop its intermediate algebra requirement for graduation. We should be excited to see a proposal from NCTM that reserves intermediate algebra concepts for elective courses in high school. But we should regard both proposals as tenuous, and understand that as people of privilege, our support should be vocal and persistent.

We can choose action or inaction here. Through your action, the definition of math may change so that it’s accessible to and enjoyed by many more people, so that many more people understand themselves to be “math people.” I want to be clear that our own privilege will diminish as a result, that we will become less special, but that humanity as a whole will flourish. Through your inaction, or through your tentative, private support for initiatives like these, the existing definition will endure, along with the existing distributions of privilege. Choose action.

2017 Nov 14. Please read a follow-up comment from Alexandra W. Logue, one of the authors of the CUNY study:

Three years after the intervention, although 17% of the traditional remedial group had graduated, 25% of the statistics group had done so (almost 50% more students). To graduate, students had to pass, not only their general education quantitative requirement (which could be satisfied by college algebra or statistics), but also their social and natural science course general education requirements. So, for many students, passing remedial algebra was not necessary in order to pass these other courses. Further, there were no differences in our results in accordance with students’ race/ethnicity. Given that Black and Hispanic students are more likely to be assessed as needing remediation, our results mean that our procedure can help close graduation rate gaps between underrepresented and other students.

86 Comments

Thank you for speaking truth to power, Dan. These ideas both keep me optimistic about the future of our little corner of the world and humble about the hard work it will take to get there.

It’s a hard thing when talking about privilege and power to write so that both those with privilege and those without privilege feel compelled by the argument. From my perspective, you do exactly that here, and that fuels my optimism that humans can rally around a new conception of mathematics teaching and learning.

Patti Scriffiny

I am in agreement with you. I was disappointed that the CCSS still seem to focus on funneling everyone to calculus. It is a great course and I love it – but I also teach stats, which is frankly way more practical. i think every student can benefit from a grounding in basic algebra when it is taught well. They can learn patterns, modeling and multiple representations of math relationships. Geometry can be a powerful and useful course as well for all learners. But what about beyond that? I teach at a rural school, where all my students are supposed to take algebra 2 or higher if they want to go to college, and if they’d rather not our options are thin. I’d love to see us offer a rich problem solving course that incorporates some discrete math, algorithms, spatial thinking, economics and other real topics of use. But I confess I feel lost in trying to create that course. Does anyone have the framework for something like this? I have teachers willing to experiment and I think I could offer it to our admin team and get it in the course catalog, but I need help. Anyone?

Dan Meyer

I’d love to see us offer a rich problem solving course that incorporates some discrete math, algorithms, spatial thinking, economics and other real topics of use. But I confess I feel lost in trying to create that course. Does anyone have the framework for something like this?

Super question, Patti. I’m really excited about Quantway and Statway from Carnegie Learning. They’re getting a lot of traction as alternatives to developmental math at the university level. I’m not sure how often they’re used at secondary but they’re worth a look.

Dan Meyer

This article came at a great time. I left my first period class today after working on factoring second degree functions and competing the square for the past week. I have not taught it procedurally but used the array model and algebra tiles and having students use questioning to work algebraically. There is still struggle however and I left today wondering what is the point.

Big.

SIRI can factor, DESMOS can do it all and most tech can now do what I’m making my students do. What is the purpose of algebra in this day and age? Am I spending time making my students irrelevant because they are learning a skill that a machine can already do?

We do so much thinking in my class and some great problem solving but then comes this pressure I feel to make sure they do can do the algebra…
I’m stuck between where I think math education is headed and what math education was.

Adam G.

Tricia, as someone who teaches math and history, I feel the same struggles. What do I have my students memorize in history class? It’s all on the internet! How much mental math do I require? They have calculators in their pockets! (something my 5th grade math teacher assured me I would not have :-)

As someone still learning to appreciate math (see my long post below), I struggle with the “why”. I enjoy factoring, I enjoy graphing linear equations, I enjoy solving algebraic equations, but what will engage my students? What will they see as relevant?

I appreciate the point made that education is more than just “you’ll use this in a job someday” and sometimes we learn something just because we can, but getting students to see that importance is a key part of their engagement. I like your question: where is math headed? If only I knew the answer . . .

Dan Meyer

Thanks for your comment, Tricia. I highlighted a paragraph that I found particularly poignant.

I just wanted to offer a follow-up thought that computers are great at some aspects of Algebra and much less good at others. They’re great at solving equations, as you mention, but they’re awful at setting up an equation from a context and they’re awful at interpreting the results of their solutions. That’s how I’d like to see “Algebra” reconfigure itself.

Jay P

Share your sentiments! It drives me crazy when students conclude that they are terrible at Math just because they cannot remember the set notation or the interval notation of the domain and range for a given problem. These are students who can otherwise analyze information and think of practical solutions using a creative approach. Even though I try to incorporate creative problem solving, when it comes to standardized tests and other assessments, students struggle, and this limits their options for majoring in Math in college. Students can’t see themselves being successful and going deep into the subject since they have faced failure early on in their school career.

While it’s true that we have tools that can do things like factoring, it’s still useful to learn how to do it. Similarly, we have calculators that can do arithmetic. But we still teach kids arithmetic, and understanding it is important for being able to think about other problems in math. Along the same lines, we can look up facts and details on Google, but we still need to know a certain amount of facts and details in history in order to make sense of new things that we learn and think about problems in history.

Whether it’s factoring, arithmetic, or historical facts, these tools shouldn’t be the end goal of education and the thing that we base all assessment on. But there is value in learning them because they are the language our kids need in order to engage effectively in problem solving.

In each case, technology provides us with a tool that can make things more efficient

This is a broader equity issue, one which transcends race, gender, social class, income level (mostly), and everything else.
So now we consider the ways in which power has been relinquished in the past in favor of the common good. Right now we have no good role models in the public eye, save Danica Roem (look up what she said about her defeated incumbent opponent).
Can we draw on the body of equity literature to highlight systemic and institutional “math-ism” (could that be the term? Nah. It’s weird.)? But that doesn’t mean the phenomenon isn’t real. It is. Is there such thing as equity and social justice for 99% of the population? Can you even refer to it as such when when the majority portion of the educated population is affected.

Consider the opportunity cost!

When I consider the time and resources spent on Algebra II and above, I think about the content in middle school skipped to achieve the goal of early algebra. I think about the broad statistical illiteracy and the quantitative and logical reasoning required to do our daily financial tasks that also are skipped. All of this eschewed in favor of the symbolic manipulation that you shared in this talk.

It’s an idea whose time has come.

What about Equitable Real Math? Nah.

Chris

“You might think, “Well that’s what math is,” but the definition of math isn’t a physical constant in the universe. It’s defined by people, just as people define the ways that wealth and power accrue in the world. That definition is then underlined, reflected, and enforced in public policy, curriculum, and syllabi.”

I think it’s hard for me because one thing you always hear about math(ematics) is how it IS a physical constant in the universe. Seems like we’ve let that belief spill over into our understanding of how math is done.

It would be worth exploring how Mathematical Platonism influences Access & Equity in mathematics education. I know this seems a bit too heady, but I honestly wonder if there’s a relationship!

I would consider myself a Platonist, along with most mathematicians. I really do think that mathematical forms occupy an immaterial space. (For example, I believe that the Pythagorean Theorem transcends time and space — it was true before it was proven, it is still true today, it will be true even after humans are gone, and it would have been true if it had never been proven at all!) As “religious” as this sounds, the experience of actually doing mathematics, self-described by professional mathematicians, is one of discovering, engaging with beauty, and pursuing truth.

How does this relate to increasing “access” for all learners? I’m curious, and I don’t have a perfect answer. But I do see our job as granting access to this eternal, immaterial space. I have pretty high and lofty goals for math education–namely, that students taste transcendence. Now it’s *really* getting religious!

Chris, you said, “I think it’s hard for me because one thing you always hear about math(ematics) is how it IS a physical constant in the universe. Seems like we’ve let that belief spill over into our understanding of how math is done.”

You describe the mathematical realist (the Platonic) version of the philosophy of mathematics, however there is an active and equally strong version of mathematics, which is the anti-realist position.

If you think of the concept of ‘infinity’ it becomes easier to discuss. The realist and the Platonist have to argue that there is a real thing, an entity, that exists called ‘infinity’ that can be pointed to, observed, and discussed in the same way a table or calculator exists. The anti-realist position argues that the idea of infinity exists only in our minds, and therefore is a construct of people.

I simplified it down to only two positions (in reality there are many subdivision of each) but in the end it shows that your statement “we’ve let that belief spill over into our understanding of how math is done” is problematic. We do math in many different ways, from many different approaches of understanding. If we don’t acknowledge the different approaches possible, then we do a disservice to our learners and ourselves as math professionals.

Daniel Knox

Love the post, Dan. As others have said, this is very timely and important!

I have to react to the comments about Statistics, though, so please indulge me for a minute.

I think pushing Statistics instead of other math classes runs the risk of missing the transcendence that math offers. Statistics rarely offers the kind of sense-making that is the hallmark of mathematics. Non-calculus-based Statistics, as it is traditionally taught in High School, has many appeals to authority and lots of blind trust in formulas. This is similar, to some extent, to the sciences, where we find math in the service of other disciplines. Statistics, although under the umbrella of mathematics, is not often taught with sense-making as the driving paradigm. The hallmark of what mathematics and mathematics education is *sense-making*.

I’d love to see the Desmos “take” on Statistics beyond scatter plots and estimation. I’m sure this is a direction you guys are already planning to go, so I’m anxious to see what you come up with!

I don’t have a perfect solution, but I don’t think putting Statistics at the center is the answer. Statistics is vital and absolutely needs to be part of our curriculum, but I just don’t think it deserves center stage.

I also don’t think putting algebraic manipulation at the center is the answer either!

John,
I agree with your concern about statistics. No one I personally know has had a sense-making experience in a statistics class, at least one brought on by the instruction. It’s as if there are formulas and interpretation of data and ne’er the twain shall meet in the classroom. But just like we’ve called attention to the sense making necessary in the mathematics classroom, the same needs to be done for the statistics classroom.
Indulge me on a literary analogy. Imagine you are a literature teacher and you make a list of 30 books that transcend race, politics, and religion and speak to the human condition in a way that is appealing to high school seniors. You can’t teach 30 books in a year, so you make really hard choices and narrow it down to three or four books. Does that take away from the other twenty-something books? Yes, but there would be no excuse, pedagogically, for teaching 30 books in a year. Now back to math. Consider transcendent mathematics like the Pythagorean Theorem. It’s on the list of 30, but the transcendent idea really is not PT, but the inherent measurement relationships that transcend human existence, pi being another example. What other ideas transcend? Which do not? Where do you draw the line? Which books do you select? Who decides?

Chester Draws

I think pushing Statistics instead of other math classes runs the risk of missing the transcendence that math offers.

Sorry, but the vast bulk of students never get any of that transcendence. Wanting it to be otherwise is not helpful. Not everyone is built for Maths, any more than everyone is built for sport.

There just is no reason to expect College students to need higher algebra, and my country (New Zealand) has no requirement for them to have it.

This is not to say that I see no value in teaching algebra, but that there is an opportunity cost — making them do Maths means they aren’t doing some other more useful subject. When someone advocates for Maths for College admission, I want to know what they are rejecting to achieve that, because these things don’t come without costs.

Is the learning abstract thinking, which is effectively what we achieve in Maths and which no other subject really gives, more important than other things. Surely a Business Major would be better off being able to properly analyse a time series graph? A Law Major will be better off having some understanding of correlation (and its relationship to causation) than what Albebra brings.

Dan Meyer

Statistics rarely offers the kind of sense-making that is the hallmark of mathematics.

Thanks, John. I think this is absolutely true and I also co-sign Kimberly’s comment above that this isn’t a function of the course rather how it’s taught. (I wouldn’t say the same about Intermediate Algebra, FWIW, which is so abstract as to defy students to make sense of it using their existing mental models.

I actually think that a non-calculus-based Statistics course really does have less *inherent* opportunities for sense-making. I’m picturing, of course, a class like AP Statistics. There are many results inside the course that cannot be proven. And if it’s a course that doesn’t include any mention of Calculus, then it seems like a crime to talk about continuous probability distributions at all!

In an Algebra or Precalculus course there are no such unproven results, except the Fundamental Theorem of Algebra (but this has a lot of intuition going for it!).

Again, the focus of the math classroom needs to be *sense-making* and I believe that very nature of the the Statistics curriculum makes this difficult, not just the teacher.

I don’t know how to make the pretty quote things, or I would quote “There are many results inside the course that cannot be proven.”

My main question is: is proving a requirement for sense-making? You mention continuous probability distributions as a thing that should never be mentioned without mentioning calculus, so I will briefly explain how I tackle the old chestnut the normal curve in my AP Statistics class. I start with the binomial distribution, which can certainly be easily understood without calculus. Then we explore “well, what happens as n gets bigger and bigger?” and of course what happens is a binomial distribution approaches a smooth curvy shape. We examine our non-continuous-but-getting-pretty-close-to-it distribution for patterns and discover a few properties. Then I say “well, this thing is approaching a type of curve called a normal curve, which has some special properties we can use to help us estimate information about the binomial curve.”

Now, at this point, I definitely do NOT introduce the formula for a normal curve (it is in the book if they want to see it and I have a couple of extra credit projects for those in calculus to do if they want) and I definitely DO get out a PDF/CDF distribution table / calculator. So from that perspective, sure, we’re not “proving” anything about a normal curve. But you know, I also teach triangle trig to 9th graders and after spending a couple of weeks building our own trig tables by actually measuring triangles we eventually get a full trig table and even a calculator and I don’t spend any time explaining how the calculator finds that number either. But they sure made some sense out of it.

Honestly, to me the BIGGEST sense-making moments I’ve ever had in class come in AP Statistics. Two things in particular are really big moments: when students wrap their brain (truly) around the idea of a p-value and when students understand what a 95% confidence interval means. These are both crazy meta-ideas that take a lot of poking around the edges and experimentation for students to really be able to grasp, but when they do it is a light bulb moment as big as any I could ever hope to achieve as a math teacher.

I have to say, your argument of “blind trust in formulas” feels far less true about statistics to me than calculus. There just aren’t that many formulas in AP Statistics, certainly not as many as there are trig derivatives to memorize. There are many amazing sense-making moments in a well-taught calculus class, obviously. But statistics is no slouch, and doesn’t require two years of algebraic work that tends to train kids OUT of sense-making by its nature.

Thank you for an extremely thoughtful, persuasive, and beautifully written piece. We now have three-year follow-up data on the CUNY study described in your post. As stated in your post: “When 907 CUNY [associate-degree] students [all assessed as needing remedial algebra] were assigned either to remedial algebra, remedial algebra and supplementary workshops, or college-level statistics and workshops, that latter group a) passed their course in greater numbers (earning credit!) and also b) accumulated more credits in later courses.” There were also indications in the data that the statistics students worked harder and enjoyed math more than did the other students.

Even more promising!

Three years after the intervention, although 17% of the traditional remedial group had graduated, 25% of the statistics group had done so (almost 50% more students). To graduate, students had to pass, not only their general education quantitative requirement (which could be satisfied by college algebra or statistics), but also their social and natural science course general education requirements. So, for many students, passing remedial algebra was not necessary in order to pass these other courses. Further, there were no differences in our results in accordance with students’ race/ethnicity. Given that Black and Hispanic students are more likely to be assessed as needing remediation, our results mean that our procedure can help close graduation rate gaps between underrepresented and other students.

We presented these results this month at the APPAM and ASHE conferences, and are preparing them for publication. We thank you again for helping to bring important issues of access and equity regarding math to a wide audience.

Dan Meyer

Adam G.

Quick personal history with math: I long considered myself one of those “I’m not a math person” people. Math was a struggle in high school and a subject to be avoided in college. I have now taught middle school math for over 10 years and am growing in my appreciation and understanding of math every year. I am becoming more of a math person every year!

I appreciate your post, Dan. I have been visiting your site for about two years now and always find good content and interesting ideas. Here is my struggle with the broader conversation of “privilege”: I struggle with the negative connotation our society now gives to privilege. I had many privileges growing up – my parents stayed married, they sent me to a private (parochial, actually – not a school for the rich, but not a public school either) school, I graduated from high school, I did not have children until I got married – these are all strong indicators of future success and things I should not be made to feel guilty about. (I don’t mean to vent here – I realize you focused on math, but I think there is another way to address your concerns).

I agree that algebra can be a barrier, but I wonder how that gap can be bridged for students of all economic backgrounds. I am working on a project for my master’s degree in education studying ways to raise achievement in my math classes. I really think math can serve to level the field and maybe statistics is the answer, I don’t know.

Sorry for the long post, and I appreciate the thoughtful post, Dan. You gave me a lot to think about (I teach history too, maybe that part of me showed through in this post :-)

Dan Meyer

I had many privileges growing up – my parents stayed married, they sent me to a private (parochial, actually – not a school for the rich, but not a public school either) school, I graduated from high school, I did not have children until I got married – these are all strong indicators of future success and things I should not be made to feel guilty about. (I don’t mean to vent here – I realize you focused on math, but I think there is another way to address your concerns).

I find it easier to think about privilege when I think about a different version of myself, someone born on the same day but just born with different skin color. That kid and I are both hard workers, and should be proud of the success that resulted from that work. But my twin will experience lots of disadvantages that I won’t, simply because I was born white and he was born black or brown.

I’m much more prosperous than he is in 2017. Maybe he also made worse choices than me, but given poverty’s effect on cognition it would be immoral of me to judge him for those choices. I’m making better choices because I’m not poor.

So I’m encouraging myself (and you!) to attribute less of my prosperity to my virtue and less of other people’s hardship to their vice. I had a lot of unearned luck and privilege.

Now we need to change laws and make reparations to ensure that luck and privilege don’t determine someone’s opportunity to prosper.

Sarah T.

This guy sounds like as racist elitist. This privilege doesn’t exist. His black twin wouldn’t be as successful? It doesn’t get anymore racist.

I came here for math tips not Marxism. Please don’t believe this garbage he is spewing. Education has been declining because it has been predominately occupied by liberals who believe in handing out everything including grades. Their test curve looks like a circle. Increasing graduation rates decrease in thinking. Keep dumbing the population down Meyer so that you can you remain the 1%.

Sue Thuma

I cannot put this blog out of my mind. I do not disagree with anything you said so why am I so upset by it? I worry about who decides which students take the intermediate algebra course because by not taking that course many occupational doors are instantly closed. We have been moving towards more equity in that regard. As a 1% member (and maybe more so as a woman) my job as a teacher was to place a hand out to pull up anyone I came in contact with in regard to taking more courses. I would think of myself as incompetent if I ever didn’t recommend taking intermediate algebra. I am still processing what you have recommended but on the surface it seems fine. Unfortunately over the course of my career I have seen students short changed mathematically due to adults deciding they could not do something. I had a graduate school prof who used to tell us if someone didn’t like math they hadn’t had a “groovy” teacher like him! I would like to think your work has made it possible for more students to succeed in intermediate algebra.

Sarah Ream

Sorry I disagree with most of this article, especially the privilege. I believe if you work hard, you will reap the benefits of that hard work. There are now students in 3rd world countries out performing us. Have we American math teachers become so defeated that we refuse to challenge our students to take intermediate Algebra. The reason we have this problem is because on paper we are increasing in rigor but in reality these students are passed along, never really mastering math. The students then go to college with unrealistic expectations based on their high school experience only to be prescribed a course load of remedial classes. Most of the students in college should not be in college. So are we to appeal to the SAT and ACT and remove intermediate math for the test to make it more accessible for students to attend college?

Woo-hoo!!! Excellent!!!! Graduation rates have increased and test scores have decreased. This has lead to students not being prepared for college.

“To begin with, let’s eliminate policies that require intermediate algebra for college study.”
“Students fail these exams in staggering numbers (68% nationally) placing into “developmental math” courses, courses which cost time and money and don’t offer credit towards graduation.
Those courses are disproportionally composed of African American and Latinx students.
Only 32% of students in developmental math ever take a math course required for graduation.”

= Less qualified people in society.

So our answer to this dilemma is to lower our standards. I am a minority and I believe that if we sell minority students on studying math they will have access to high demand fields where its easier for them to enter the workforce and have a lucrative career especially since the don’t have “privilege.” (Which I think is a made up concept and sounds elitist in itself). I teach my students that there is no limit to what they can accomplish. What happened to “Stand and deliver?” Minorities are capable of learning math if they receive an adequate education and are not passed along. Minorities from other countries including 3rd world countries are surpassing Americans. Let’s learn from Ben Carson, Mae Carol Jemison, Clarence Thomas etc. Lets challenge all of our students to master math. This will build perseverance and character. Let’s learn from my family that immigrated from the dirt and poverty of Haiti to become teachers, doctors, lawyers, engineers etc. in one generation. Stop selling minorities short. I didn’t need standards lowered for me. I needed teachers who believed in me and cared about me my future. When my counselor tried to steer me to a community college I was able to get admitted into the top 3 colleges in Florida with AP Calculus college credit.

I know you are tired of the friction of teaching under prepared disgruntled students, hearing “Why do I have to study math?,” and feeling tired but you will produce a generation of hard workers.

By the way, for the person struggling with Latinx it’s the PC term for Latinos and Latinas.

Adam G

Sarah, you articulated what I was trying to say in an earlier post far better than I did. There is so much of your post I would quote in agreement, but I’ll just summarize your point about hard work and our failure to hold our students to a higher standard.

But what should we force students to learn.

I echo your words to my students too – they can learn anything!

That’s what I was getting at when I explained my own math journey – struggling in high school and now a math teacher (even Algebra 1 some years).

“Lets challenge all of our students to master math. This will build perseverance and character . . . I didn’t need standards lowered for me. I needed teachers who believed in me and cared about me my future.” I agree completely. Thanks for giving me some things to think about and make my own argument better next time.

I hope you have a great school year challenging your students and helping them surpass their own expectations of themselves.

Sue Thuma

Oh, my! I just saw your tweet about this article and I just realized you are talking about college. It was late when I read the blog! My remarks above were referring to High School Algebra 2nd course. If Algebra 2 were well done in high school then we wouldn’t be having this discussion because we would be passing that college entrance exam. If a failing student of that pretest didn’t have to take the test at all would they still fail to graduate? We will never know.

Dick Fuller

Established education is responsible for the idea that mathematics is what is convenient to test. That’s what makes it a great gateway. Statistics offers the same convenience, but what if learning to understand is really important. How does stuff work, not merely how to work it. What does it mean to really understand?
If mathematics is taught for “quantitative reasoning” by all means replace it by statistics. Your students will face a certification culture, they should be prepared.

Isn’t there a deeper problem here? From the beginning, meaning in school mathematics is what the instructor, text book, standards says some association of a mathematical object with a physical one says it is? Think about the mess made with negative numbers and fractions. Then it is necessary to teach arithmetic as rules and tricks. There is no mathematics available. What’s a student supposed to take away?

Teaching owns the truth, and the truth is what education owns. It is not mine to be discovered and understood. But I must yield to this authority of instruction, because I see no other way. They’re going to test me on what they say math is. It is just like all the other imposed authority in my life. I must learn to fear it. I hate math.

I agree, Sue. Dan said that computers are “great at solving equations…but they’re awful at setting up an equation from a context and they’re awful at interpreting the results of their solutions. That’s how I’d like to see ‘Algebra’ reconfigure itself.” For the most part, the CCSS did that. The problem is that very few understand the intent of CCSS Algebra 2. Student Achievement Partners (Dr. McCallum’s non-profit) reposted this piece recently to explain the whole Algebra 2, STEM, college disconnect: https://achievethecore.org/aligned/8-questions-about-high-school-math-and-stem/

As a generally misunderstood ed-tech entrepreneur, I have so far found math teachers to be very insular. Anytime I can even actually beg one to talk to one, it ends up in a discussion of how they hate ed-tech, specially if its not built by a math teacher. I almost never get to actually discuss the merits or demerits of what I actually do, and how we may collaborate. I find them endlessly repeating the same echo chamber of their specific 1%, with little to no desire to incorporate anything real or realistic into their teaching or learning. Your examples, while lamenting the math 1%, still use abstract examples of geometry without connecting them to anything real and are still stuck in the same echo chamber. If you want to make your math teaching real, you may have to get out of the classroom (and I don’t mean to the grocery store line) and talk to people who use math every day: in construction, in design, in biology, in physics, and yes, with computers, robotics, even games too, or at the very least be open to understanding these uses. Till you can make the connection of math to its very real uses, you’ll be blissfully, ignorantly stuck in your echo chamber, and be unable to inspire your students.

Dan Meyer

Nils H

Ana, some would say that we don’t ask artists to justify their art by citing when it will be used in real life. Neither do we do this for the study of literature or history. I’d even venture that your interest in computers and tech didn’t start from utility but rather curiosity and challenge. If utility made subjects interesting, I have a wonderful course of The Fork in development.

Rather, the challenge and curiosity that have been drained from Math needs restoration. My Stats students are not nearly as interested in formulas as they are in trying to create response bias and how to test it. Google One Cut Challenge and watch middle schoolers try and try again to make specific shapes with only one cut of a piece of paper by folding the paper. A bell cannot stop that fun and no one is asking where they will use it.

paul martin

I have some sympathy with your view. Over here in Scotland the focus of our education system is refocusing on the basics of Math & literacy. At secondary school the emphasis in early years is attaining a common level of comprehension. This is subsequently tested in a range of exams at Higher & “National” level (in reverse age order). As an invigilator I have seen the focus oscillate, with a general direction of more practical math. Last year I had a chance to chat to a math teacher about the SQA Higher Math Croc question controversy:

I congratulated him that the students taking the exam, in the previous diet, which I supervised were calm and had no emotional outburst as reported elsewhere nationally. His answer was “but none of them passed”.

When I was at University Math (& Psychology) were both sources of students for other courses as the first year exams were designed to weed out the weak – those who did not transcendentally comprehend maybe. I have enjoyed watching my daughter bounce back after moving on from medicine to achieve a math degree from the excellent distance learning provider Open University, saving enough from tutoring and now enrolled in a Big Data postgrad stats course in London.

Ana Redmond

OK. Here’s an example. The polygon->circle example you presented has very real implications in game development. In 3d OpenGL, everything is rendered as triangles. Makes it hard to do circles. Older games used jagged polygons instead of circles, since that’s all that could be rendered with those computers. As the capability of computers has increased to render more and more polygon count, the curve has smoothened to look like a circle to the eye. In complex games, we worry about polygon count. If there are things we can represent with less triangles (and by extension polygons), its computationally cheaper. These days we often use software that handles the polygon creation for us. But, in the beginning they were done by hand. There is a very famous image of the graphic modeling of a teapot: https://www.sjbaker.org/wiki/index.php?title=The_History_of_The_Teapot Stories like this fascinate me to dig deeper, ask questions like: why use polygons/triangles? What if we changed graphics cards so it rendered curves to begin with? What does a curve mean in 3d space? What is the best polygon count on a low-res device vs. a high-res device? I don’t know the answers to these questions. Its just that the math is now far more interesting to me.

You talked a bit about the math epiphany as your primary motivation for pursuing this career, but then, if we were to design a curriculum system based on it we would have to fundamentally overhaul the system so that it doesn’t optimize on practicality.

To be sure, there are schools out there that are structured that way, but it’s unclear if this model should be applied to all. As deplore as the verify-compute-recall model is, they are there because they appear to the shortest path to get to the mechanics given the strand resource we have. Maybe equality is not so much of an issue as much as diversity is. And that’s why we still have both applied and pure mathematics standing hand in hand without one dominating another.

Rich Konarski

Disagree about Statistics. Data is where the lines, quadratics, exponentials come from. It’s visual which allows students to discover patterns and analyze them. Not only that, stats is apart of every class (grades, probability, percentiles, reading graphs, the scientific method…). Plus, almost every college major requires some sort of statistics, just ask a nurse how difficult stat was. Stats before Calculus, I say. Imagine knowing what a normal distribution is then diving deeper into understanding integrals and the smaller and smaller rectangle thing.

I don’t really understand why statistics should be a better course than algebra for college graduation. Couldn’t we make a really technical and difficult statistics class? Is statistics inherently easier than algebra?

So why is statistics a good replacement for algebra? Is it just because it’s easier and less technical? Would an easier and less technical algebra class have the same benefits?

I’d guess that for THIS STUDY the results have to do with the fact that statistics was probably completely new to the students so they were able to approach it with fresh eyes, the professors weren’t thinking of their students as “students who already failed this material once.” I expect that if statistics replaced algebra at all levels (high school AND college) then this effect might fade, since now the entrance exam and the remedial non-credit courses would themselves be stats classes rather than algebra ones.

I personally think that statistics makes more sense than intermediate algebra (for the nth time) as a college requirement because it has somewhat more practical applications for students who are simply trying to make it through the minimum math requirement. Being able to have at least a vague understanding of a p-value, confidence intervals, what “r=0.5” means, and that when the fivethirtyeight.com model gives Trump a 30% chance of winning that doesn’t mean they “got it wrong” when he does when is imminently more useful for the vast majority of college graduates than the algebra examples Daniel shows. And you can get a lot of good mathematical thinking out of it too, if the course is taught well. But I do think that you’re right it probably doesn’t ITSELF make all or even most of the difference found in this study – too many confounding variables (ha! Statistics!)

Noah

Perhaps the CUNY study needs another treatment group: students who take neither the remedial algebra course nor the statistics course. Maybe those students would fare just as well as the statistics treatment group in their subsequent academic work. This seems plausible to me given some of the literature on social promotion.

I’d like to add some research context to this discussion of math remediation and algebra vs. stats in college (more info. & cites can be found in our paper about our experiment at http://bit.ly/2mKv9Y4). First, the majority of new freshmen in the U.S. are assessed as needing remediation in math, writing, or reading. Math (i.e. algebra) is by far the most commonly assessed need, and most students never take or do not pass their required remedial math courses. Therefore math remediation has been called the single largest academic block to graduation in the U.S. (and note that students not graduating is the biggest reason for student loan debt default). Now you could say that students who get into this situation (and that applies more often to students from underrepresented racial/ethnic and economic groups) can’t do the work & shouldn’t be in college, but data show that in many cases that statement would not be accurate. Approximately 25% of students in urban colleges assessed as needing remedial math using tests could instead be put into college-level math (with no extra help) and receive at least a B (Scott-Clayton & Stacey, 2015, http://bit.ly/2A1CFbK). The placement tests make a lot of errors, including because how motivated students are (vs. their preparation or abilities) is extremely important to how they do in a course, and repeating a high school math course for no college credit (and using your limited financial aid to pay for it) is not motivating. Further, while students from challenging backgrounds are completing their remedial sequences, the longer those sequences are, the more likely the students’ difficult lives are to intervene, so combining remedial levels or using corequisite remediation (as we did in our experiment) decreases the opportunities for students to exit remediation (for reasons beyond their control) and not complete the sequence. The third aspect of math remediation reform (in addition to placement and acceleration techniques), and the most relevant to this discussion, is the kind of math/quantitative work that the student does. You can say that every student should be required to know algebra, but if you do that for students who aren’t in algebra-intensive majors, then a huge percentage of these students will never finish college due to not passing their math courses, even though they will never use again the huge majority of the material in those courses (Douglas & Attewell, 2017), courses that will not benefit them in their post-college earnings (except for women who take stats in college, see Belfield & Liu, 2015). Alternatively, we can give students rigorous statistics or quantitative reasoning courses that contain material more suited to their interests, majors, and career goals. As our experiment with students assessed as needing remedial algebra has shown, we can instead give such students college-level stats with extra support and they will be more likely to pass the course, to accumulate credits in the subsequent year, and to graduate (which includes passing their natural and social science gen ed required courses, even though they never took the remedial algebra they supposedly needed). Finally, note that it is not possible to say that a course in one subject (e.g. stats) is “easier” than another (e.g. algebra), because they are qualitatively different courses and there is no one easiness scale on which you can locate them. What you can do is make sure that there are clear learning outcomes for every course and that students are held to those outcomes, no matter by what path a student came to that course (as was done in our experiment). In conclusion, in a country that in the past 20 years has fallen to 13th in the world in terms of young adults with college degrees, and where the need for people with college degrees is increasingly outstripping the supply, there is another approach to quantitative literacy for college students other than algebra for all, and it is an approach that helps us to provide equitable higher education opportunities for students from all racial/ethnic/economic backgrounds, while also helping all students live future productive lives.

Thank you for the research context! Here’s my attempt to summarize/synthesize:

(1) Remedial algebra is demotivating for a bunch of reasons. (No credit, it’s a repeat.) If you swap remedial algebra with something more motivating (credit, new material) more people will pass it and graduate. This is good.

(2) Because remedial algebra is demotivating and offers no credit for graduation, it extends the time needed to graduate and makes it more likely that something — e.g. a need to work, a need to care for a child — will force the student to drop-out. A more motivating course that counts towards graduation would help more students graduate.

(3) All things being equal, statistics is more interesting and relevant to students than algebra.

(4) You can’t compare the difficulty of an algebra class to a statistics class, so we can’t say which is more difficult for students.

—

If this is a good summary, then I totally understand (1) and (2), but I’m not at all sure about (3) and (4). It seems to me that there is nothing inherent about statistics that is more interesting to people than algebra. It’s all about how you teach it.

It also seems to me that there’s nothing more relevant about statistics. Maybe I’m delusional, but it seems to me that you can go about your life just fine without statistics, just as well as you can avoid algebra in your daily life as a citizen.

(Both algebra and statistics — and music and philosophy and poetry — can enrich your life and your ability to critically evaluate your society. I don’t think statistics has some special status.)

So I’m not sold on (3) — I don’t think statistics is just more interesting/relevant than algebra.

And I’m also not sold on (4). It seems to me that it’s entirely possible to assess the difficulty or ease of a class based on crude things like the completion rate. Ease and difficulty is, as you say, not a natural thing, and it’s difficult to disentangle difficulty from interest.

Because I buy (1) and (2) I’m convinced that we should not require remedial algebra. We should replace it with something that is not a repeat of high school math and that counts for your graduation requirements, or at the very least doesn’t slow you down in your path to graduation.

But why should that class be statistics instead of ____________? It seems to me that we’re saying that statistics classes are INHERENTLY easier for students to pass than algebra classes. And if that’s the case then it makes sense I get the argument for swapping algebra for statistics. But this doesn’t sound right to me. We get to decide what “college-level” statistics is, and we can raise or lower those standards so that, in aggregate, we can pass or fail any percentage of students that we’d like to.

I think what’s really happening here is that we’re trying to pull a fast one on our unjust system. We’re saying, let’s swap out statistics for algebra. We’ll tell everybody that the statistics classes are high-level and while that won’t be a lie, it’s going to be easy enough that many students who would fail remedial algebra will now pass statistics. And in the short term that will be a good thing, because we’ll have gotten a lot more people to graduate. In the long term, the statistics classes will just recalibrate to whatever our current status quo is.

(Or maybe something unexpected will happen to make us change how we think about math for graduation.)

So from my point of view, what should we do? Like I said before, addressing (1) and (2) makes a lot of sense to me. Remedial algebra does not make sense to me. It sounds like our argument should be that remedial algebra needs to be either cut or replaced with a course that is much easier to pass, as it doesn’t really do anything except make people drop out of college at a higher rate, and for all the wrong, unjust reasons. Remedial algebra has got to go.

I would just hate if statistics got caught up in all this, as it seems sort of a distraction to me. The point is not that statistics (or coding, or “quantitative reasoning”, or anything else) is inherently better for students than algebra. It’s just about finding a way to sell higher passing rates to the system.

Michael makes an excellent point — is swapping algebra for stats just a proxy for “swapping something difficult with something easier”?

Many people are bringing the criticism that we’re lowering standards by making this switch (see many of these comments, including Sarah Ream’s, especially). Are these courses “equally rigorous”? We really need to address this question if we’re going to respond to these charges.

This is addressed to Michael Pershan’s comments on my comments. Regarding #3, I agree that a lot of this has to do with the way a course is taught, not the course itself. What the research seems to be showing is that if a math course is taught so that students can see the concurrent use of the material outside of that classroom, they do better. Statistics is usually taught with lots of applications; remedial algebra may be less so. However, when the parts of remedial algebra needed to understand intro chem are taught together with the intro chem (instead of in a separate, preceding class), i.e., when the remedial math is taught as a corequisite vs. as a prerequisite, the students do much better with the math (see Hesser and Gregory, 2016, in Journal of Developmental Education). At CUNY we’ve had similar success taking this approach also with intro sociology and nutrition. Some students need algebra for their majors/career paths, and are not yet prepared for college-level algebra, and they should have remedial algebra, but taught in the most effective way possible (which the data show is as a corequisite). Other students have a greater need for stats (e.g., recall the Belfield & Liu cite in my previous response–this paper shows that only for women taking stats in college does what math you take in college help post-graduation earnings), and others for a course in quantitative reasoning. What sort of math a student takes should be tailored to each student’s interests and needs (The Dana Center at U Texas Austin has done a great job exploring & explaining this, please see their website at https://dcmathpathways.org/?q=%2F). Regarding #4, I’m not sure what you mean by “completion rates” (completing the course? a degree?) but regardless I don’t believe completion rates will tell you whether stats or algebra is “easier.” Specifically looking at course pass rates, an instructor has all kinds of ways to make them go down or up (many of these ways being educationally questionable or worse). And some instructors are indeed better than others. So course pass rates by themselves don’t say whether the subject itself is easy or not (whatever that means). Going back to my response concerning #3, I think the way the research is coalescing is that, for many students, what helps them learn faster is seeing the applications of what they are learning, whether that be statistics or algebra or QR, but there is probably not yet enough data to say that for sure, and it may not apply to all students.

What sort of math a student takes should be tailored to each student’s interests and needs.

Absolutely. This seems like the heart of the matter. Remedial algebra should be killed and replaced with sensible math requirements that are co-requisite to their other majors.

But I took the post to be suggesting something more — that we’d be better off swapping remedial algebra with a statistics class, as a default.

And, looking back at all the things I agree and understand from this conversation, I think I’m compelled to agree with you. After all, the material is new and the course wouldn’t be remedial. That would be less dispiriting, more motivating, and as a consequence more students would likely pass it…

…but part of me is nervous about the swapping-out argument for statistics still. Ultimately, most students don’t need statistics either. So the only benefit for statistics vs. algebra for remedial students is that more students, for whatever reasons, pass those statistics classes than those who pass the remedial algebra classes. I find thinking about this to be very confusing. There is no objective standards for “college-level statistics” and no inherent passing rate. Aren’t there LOTS of ways to change remedial algebra so that more students pass it? (There’s nothing inherent about “college-level algebra.”) I find this very confusing.

I don’t mind this at all.

But I don’t find something like the Dana Center’s project confusing at all. The point is that it doesn’t benefit society in any way to have arbitrary roadblocks on the way to college graduation. Math should be required for majors in which it’s necessary, and should be available for students who are interested in it. That seems like the point. The swapping argument is confusing to me and I think it’s unnecessary to this overall project.

Sue Thuma

My point above is that IF we are talking about high school, not taking Algebra 2 in high school AUTOMATICALLY makes a student fail a mathematics placement test should they decide to attend college at a later time. I am not comfortable telling a high school sophomore NOT to take Algebra 2 thus condemning them potentially to low paying jobs forever. A college Algebra 2 equivalent is the minimum credit producer at most colleges. I believe Algebra 2 is where most students actually learn Algebra 1! Dan’s point that some majors do not need math is well taken but to be fair in high school we don’t know who those people are and neither do they. One of my former colleagues used to tell her students “You don’t know where you are going so pack heavy.”!

Nils Headley

Uma, Algebra 2 can still be taken before college. There are many branches that can be studied from graph theory to probability to counting systems (see James Tanton) to cryptography to good old problem solving to the history of Math. When a college placement test has 10 questions on function notation, we have a problem with what education sees as Math.

Dan Meyer

My point above is that IF we are talking about high school, not taking Algebra 2 in high school AUTOMATICALLY makes a student fail a mathematics placement test should they decide to attend college at a later time. I am not comfortable telling a high school sophomore NOT to take Algebra 2 thus condemning them potentially to low paying jobs forever. A college Algebra 2 equivalent is the minimum credit producer at most colleges.

This is a valuable concern. If every college required Intermediate Algebra for entrance, I’d worry about rescinding Intermediate Algebra for high school graduation. But I’m urging every college to reconsider the math they require for graduation. The California State University no longer requires IA for graduation and it’s the largest four-year public university system in the United States. That expands possibilities for high school students rather than shrinks them.

Concerned Citizen

Not everyone belongs in college, and to reflect this, admission requirements should be much tighter. Not everyone who gets into college needs to graduate; what’s more important is that the students who do, have learned a great deal and gone through challenging courses. When the market is un-flooded, and the ‘education industry’ is de-commodified, problems like needing a degree for a job, and like an unsavoury system of standardized testing and standardized tutoring will vanish, in time.

And it will take time, because men like you have been doing your nasty termite’s work for decades. It will take time, because men like you have, with malice aforethought, lowered our standards and degraded our institutions to the point where simplifying, what was it? Simplifying 2/3x – 1/x is just TOO ABSTRACT for over 60% of your postsecondary students. That’s sixty percent of our highest-achieving 17- and 18-year-olds, so they are legally adults, and they are failing miserably at a simple children’s task. And from this you conclude that universities are still too meritocratic, or in your words ‘configured to preserve mathematical privilege’.

If you are serious about helping people learn secondary-school mathematics, then by all means, go out and teach this material to students, in a secondary school, or in an equivalent after-hours school for adults. Please, PLEASE leave the higher education alone–your kind have done more than enough damage already.

Nils Headley

1) If college doesn’t sort, there’s no point to college. If college doesn’t need algebra, why does the job need college? Lowering the standards for college will simply–and has already–devalued a college degree. And this is already happening in two ways. First, some kids are going through higher and higher math in high school (way higher than is credible they can master), so that they can get into more demanding colleges. Second, some kids are finishing college while in high school. The first group are kids of privilege. The second group are underprivileged, low skilled kids. The “middle college” option was created for low-skilled kids to keep them motivated while in high school, so they are getting college credit just for going to high school.

So we are taking kids who are lower skilled and less motivated and saying “hey, let’s call what they’re doing college work so they can be more motivated and we have a chance of giving them more skills!” Meanwhile, kids who are actually skilled would never dream of entering that program because they can *really* do college level work, so they’ll stay in high school doing much harder work than the kids in the dual credit or middle college option. No one is fooled by this. Employers aren’t going to think “wow, this kid went to college earlier than that kid, so he must be much better!”

The entire point of college, originally, was creating a more demanding course set that not everyone can pass. If we are abandoning this mindset, then it’s worth asking what the point of college is.

In no way am I saying this is about hard work and commitment. I’m merely pointing out what David Labaree has already observed about our school system: Someone has to fail. All these well-meaning people looking for ways to push more kids through college are ignoring that reality. There will always be distinguishing factors, something that will sort. For the past decade or so, we have engaged in shocking (to me) level of lowering standards for college in the utterly misguided belief that giving kids low-value college degrees will improve outcomes. This is making it harder for the lower middle class kids and the genuinely able low income kids who are the ones going to colleges that are dramatically lowering standards by ending remediation. It’s not doing much to improve life for low income kids. My preference: a job market that allows kids to ignore all this happy talk about college. But that’s a different debate.

2) As long as we’re saying “hey, middle school math in college is fine! Plus stats!” then why the HELL are ALL high school kids required to pretend to go through two years of algebra and geometry? They clearly aren’t learning it. At least three state college systems (CSU, CUNY, Tennessee) are ending the entire notion of remedial math. They will now give credit for courses that are basically middle school math. But they are *still* requiring that their matriculating students show Algebra 2 on their transcripts.

So colleges are saying yes, we know you can’t do math. That’s ok. But you better have algebra 2 on your transcripts! And at the high school level, no one seems to be thinking about privilege. Anyone who observes that, perhaps, not all kids are ready for algebra one, much less algebra 2, in high school, is shouted down. But if we were in fact allowed to sort kids by ability to learn advanced math, if we’re allowed to acknowledge that college doesn’t in fact require calculus, then we could perhaps better prepare the lower ability students for college, which is now just fine with middle school math.

We’re beating up high school teachers for not getting all kids to equal levels of success. But if it were actually high school teachers’ fault, these same kids would be learning the higher level math in college. They aren’t, clearly, or we wouldn’t be engaging in this debate about hey, who really needs intermediate algebra.

Meanwhile, no one’s beating up colleges. Instead, at the *same* time we’re pushing more demands on high school students and castigating K-12 for not meeting these demands, we’re praising colleges for “increasing access”.

The upshot: we aren’t allowed to acknowledge that a big chunk of kids could get free lower level math instruction in high school. Instead, we force them to take advanced math they don’t understand, and pass them because otherwise the schools would be at risk for failing kids (usually of color). So they waste two to three years in high school not understanding math. But if we can keep them motivated–that is, give them passing grades and reduce curriculum demands–then we can get them to college, where they won’t be expected to live up to their transcript. If we hold them to higher standards, they’ll get discouraged and get low grades and thus not be allowed to go to college. This is just increasing the need for high school teachers to engage in active fraud.

I don’t know when our country will stop confusing the credential with achievement, but I hope we get there soon.

Chester Draws

Well, most jobs don’t need college. People need college degrees to prove to employers that they are worth employing, not for the skills gained as such. Almost every job, would be better off being taught vocationally.

Even Maths teacher. Most of my degree is a complete waste of time in terms of being useful as a teacher, since there is no way I need all the abstract Algebra I learned to teach at high school. A three year course mixing technical subject areas, but not to third year university, and practical teaching experience would give better teachers in a shorter and less expensive time.

Yes, we learn other things at University than our specific subject area, and that’s good in its own right. I’m just pointing out that almost no jobs take what we learn at college and apply them directly.

Personally, I suspect that most college degrees are a total waste of time. My Masters study certainly was.

Dan Meyer

Even if we grant the premise, I’m not arguing that college shouldn’t sort. I’m arguing that college should change the criteria on which it sorts.

As long as we’re saying “hey, middle school math in college is fine! Plus stats!” then why the HELL are ALL high school kids required to pretend to go through two years of algebra and geometry?

I’m not saying this. But I am saying “middle school math, algebra, geometry, and stats are a) useful for many professions, and b) even if they aren’t useful for your profession, they’re useful for a general education.” The only reason I didn’t make this point explicitly in my talk is that the audience was primarily composed of university faculty, not high school teachers and administrators.

Scott Farrand

I am not excited that the CSU has dropped its expectation that students should have learned some algebra in high school, but it might not be for the reason you expect. I believe that the version of algebra that is commonly taught amounts to some sort of ritualistic abuse that we apply to the population, apparently to rid many students of any enthusiasm that they might have for learning. And I agree that for the majority of students entering college, the algebra they have or have not learned is not relevant to the rest of their lives. But for students headed for quantitative majors, it is very relevant.

I’m not arguing with you, Dan. Rather, I am trying to make a point about this policy change that needs some serious thought. Any frustration you find in my words is with the CSU system, and how it has ham-handedly dropped this on California, all too suddenly.

I am concerned that this radical change in policy will shut the door on many students who grow up in poor neighborhoods and want to be STEM majors. Currently, they are likely to attend a high school where the teachers are working devotedly to assist students in getting into the CSU. A student who is accepted into the CSU is a huge success for these teachers. They know that they need to get the students to some level of success in algebra and geometry, because that is part of the entrance requirements. By removing any real expectation that students have learned some algebra and geometry in high school, the CSU has clearly indicated that the high schools can lower their expectations in these subjects. Less algebra will be learned, but more students will reach college. As you point out, this doesn’t really matter to most students, but to a student who wants to be a chemist, or an engineer, or an economist, or … it matters, because they lose some opportunity to learn, in their high schools.

Schools in wealthier neighborhoods have enough students going into the UC system, so the standards for their coursework seem much less likely to be impacted.

What happens to the student who does her best in high school, and wants to be an engineer, but didn’t really have access to coursework in high school that would prepare her for calculus in college? She can still get into the CSU, but she is likely to need even more remediation. This comes at the same time that the CSU is engaging in magical thinking, believing that we can just wish away the need for remediation and declare students ready for higher level coursework. She will find much less remediation available, and she seems even more likely to move to non-STEM major.

Anyone who cares about issues such as access to STEM careers for ethnicities that are historically underrepresented in STEM should think about the effects that this policy change might have. For example, how will this impact the numbers of minority students who become math teachers?

I can agree that this policy change will open the door to college for many students who are in fact quite bright but have been held back by their inability to make sense of the mathematics that they have been taught, and that this might have very positive effects on students from low-income families. I will not argue against this change, but I do argue that the effect of this policy on future STEM majors needs to be front and center in our thinking. So far, I have heard almost nothing on this subject coming from the CSU. If we aren’t thinking about this now, then when? We can’t allow underrepresented students who might otherwise have STEM careers to be an afterthought.

Sarah T.

Dani P.

At first read, I couldn’t understand why any high school or college wouldn’t require Algebra 2 to graduate. This is probably attributed to me being apart of the mathematical 1%. Just like your mathematical epiphany of a polygon inside of a circle, I have been lucky enough to have many of those throughout my lifetime and I couldn’t imagine a world in which we deprive students of these epiphanies.

As I continued to reflect though, I begin to agree with much of what you were getting at. I received a dual major in Businesses and Actuarial Mathematics from the University of Michigan.

There it is.

Calculus I was a requirement to apply to the business school. Although, as a math major, it wasn’t an issue for me, but it was for many of my fellow classmates. Now as many as you might guess, as I went through my 3 years in the business program, I never used calculus once. Throughout those three years, ONE teacher mentioned calculus ONCE. Her words I’ll never forget: “This model can be solved using calculus, and I know it was a requirement to get in, but we won’t use it in this class.” If calculus was a requirement, then why didn’t we ever use it throughout the program? In my opinion, it was just a way of sorting and figuring out who “had the skills” to be apart of the mathematical 1%.

I put “had the skills” in quotes because I believe any math class should teach students more than just the math concepts, but problem solving skills. As tracking occurs, these problem solving skills are often lost at the lower levels and highly valued at the upper levels. As long as students learn problem solving skills, I would be find with them not learning anything beyond Pre-Algebra. To be honest, in most careers you don’t need math skills, as there is technology to do it all for you. However, what is need in all careers, are the skills to think out of the box, apply what you know, and solve a problem.

Dick Fuller

Dani P.

I agree that “math concepts” and “problem solving skills” are inextricable for the mathematical 1%. However, when you are requiring students to take certain levels of math to graduate, the problem solving skills tend to disappear and rote memorization in often implemented just to get the students to pass. If lowering the “math concepts” required for graduation meant more problem solving skills would be taught because there would be more time in the classes to explore a topic, then I would be in favor of it.

Dan Meyer

Thanks for the provocative conversation, most of you! I certainly commend to your attention comments by Alexandra Logue, a member of the CUNY research team I mentioned in the post; Michael Pershan, who spins the issue down thoroughly into constituent parts; and Scott Farrand, a CSU professor directly affected by this policy proposal on intermediate algebra.

Here are some of your objections to ending intermediate algebra as a requirement for college (and high school!) graduation, along with my responses.

You’re saying minorities can’t learn.

The fact that remedial algebra courses are disproportionally composed of African American and Latinx students is not evidence that African American and Latinx students can’t learn as much as other racial subgroups. (Indeed, no such evidence exists.) Rather, it’s evidence that those groups have been systematically deprived of the opportunity to learn, and adds force to proposals that counter that deprivation.

So you’re saying we should lower the standards.

I’m saying we should predicate college graduation only on knowledge that is useful for a) the professional education or b) the general education of a very broad subset of graduates. That says nothing about the difficulty, kind, or number of courses required. If we’re very concerned about diluting the value of a diploma (I’m not!) I’m open to replacing Intermediate Algebra, a course that fails both criteria, with multiple courses that fit the criteria instead.

But they won’t be able to get high-paying STEM jobs without Intermediate Algebra.

True. Nothing about this proposal explicitly prevents students from taking IA, though. This proposal would redefine the floor for college graduation, not the ceiling. That said, I think Scott Farrand and Sue Thuma and others are correct to point out that, as a result of this policy change, high schools may dedicate fewer resources to IA, which may penalize students at underresourced schools who want to take IA.

Those are speculative damages, though. and students are suffering actual damages. The proposal will benefit students who don’t want to take IA, students whose pursuit of good jobs in health, education, the social sciences, even business, stalls on a course they don’t care about or need, students who drop out with debt and without a degree.

It’s the unearned privilege of the mathematical 1% to say, “Even still, you all need to clear this hurdle anyway.” That needs to change.

Also, I have to point out that while many of the commenters here have argued for IA’s value for professional work, not a single one of you has argued that it’s an interesting course, one worth of general education. Farrand describes it generally as “ritualistic abuse.”

How is it any kind of enticement into STEM fields to require students to take a course that even this crowd of mathematical elites dislikes?

Sarah T

Dan Meyer

Latin professors love Latin and bee keepers love keeping bees. That doesn’t mean we should require either for college graduation.

I’ve listed my criteria for those requirements. I’m interested in alternates. But “some high-paying jobs are available” and “some teachers love it” isn’t sufficient for the 32% of students who successfully matriculate from developmental math.

Mary Gentry

Don’t disagree with you, but want to add a wrinkle: decision-makers outside of the math field who use passing math course “x” as a litmus test (or predictor) of success in other courses often in and/or out of math. In my school, AP Calculus is prized over AP Statistics simply because of how it looks on a college application. If we wish to have real conversations about the mathematical experience of students, the math gatekeepers need to be at the table as well.

Dan Meyer

Agree. These decisions and reforms will involve everybody because they touch everybody. I’m happy to see the CSU provosts doing their part. Teachers need to also realize their power as advocates for these reforms, though. As gatekeepers themselves. Their silence will be interpreted as a defense of the status quo.

Valerie Koch

As an international teacher, it seems to me that Americans are singularly focused on the importance of algebraic understanding of math with an eye toward calculus. Looking at other countries’ math curricula, you’ll see a much broader based understanding of what is important. Algebra, yes. But also a strong emphasis on statistics and probability, logic, patterns, sequences, geo and trig. I am for other math options for kids, other ways to show their ability to reason and logic, other programs that may empower them to actually be successful in life and in work.

Daniel

I’m a relatively young teacher (4th year), and I have come to this conclusion, as well. I am our AP Calculus teaching, so one would think I support a universal Alg 2. However, Alg 2 is a “powerful” course for a small percentage of our high school students, yet we require it from all of them.
I also strongly dislike when people say, “This math course is important because it teaches you to problem-solve”. Sure, yes it does. But we should be teaching relevant mathematics, not simply problem solving. Why not have both?

I think I can guess why statistics has a higher pass rate, and it’s not just that being a for-credit class is inherently more motivating than being a remedial class. When I taught something akin to Algebra 2, I found that students did much better in the Trig portion than the Algebra portion, simply because Trig gave them a fresh start: their persistent misconceptions about algebraic manipulation would only hamper them in about 10-15% of problems (like identities). In the rest of the subject, they got to start math over.

I’d guess something similar is going on with Stats.

Sorry if this was mentioned before. I tried to read the whole thread, but I admit so skimming some of the comments.

dtfitzp

I’m really new to this conversation. I recently attended a Missouri Math Pathways Symposium and Amy Getz from the Dana Center at UT Austin (http://www.utdanacenter.org/about-us/staff/amy-getz/), characterized “College Algebra” as a Pre-Calculus class, from which only 10% of students ever go on to take Calculus. Now, why on earth would we require everyone to take a prerequisite course for a class to most students will never need. If we want transendence someone should teach, “Math that will blow your mind.” If you want competent consumers of information you should teach “How statistics are made and interpreted.” If you want good math teachers you should teach “Understanding how math works so you can understand what your students are doing.”

NMB

I have been teaching IA for approximately 20 years and have always began the course with, “ We will learn math that most of you will not use in your life except to improve your ACT and SAT scores. “ I agree that this course should not be needed for college admittance, but the change must come from the top. Public education is left up to each state, so as a country we will never be at the same page.

NMB

I hope we don’t get caught up swapping one course title for another and ignoring what we really want to address. Comparing summarize-analyze-predict to slopes-and-areas-and-optimizing: either can follow intuitions and be made applicable. Or either can plan too little or too much or get caught up on esoteric details. All those who cited MM Tai think Calculus was super interesting and useful, and a bunch of med school journals about instant cups of soup include a fair amount of Calculus.

Students learn Algebra from their teacher; outside influences don’t influence what a student knows. Were we to replace Algebra with some other logical system which is considered more immediately applicable, then knowledge of other areas or media or friends and family would give some students advantages over others.

I imagine that the Calculus requirement came from a place of expecting students to demonstrate conquering a course sequence decided at the time when a course sequence in Statistics wasn’t invented yet; what would that sequence look like, and would it end up with the same complaints?

William Carey

When people propose a radical change, I’m always interested in how things got to be the way they are. Why is intermediate algebra a requirement for college graduation? Why is is central to students high school experience? I’d be really wary of removing it without understanding exactly why it’s there in the first place.

I’d propose that what unifies intermediate algebra is the idea of a truth-preserving transformation. That we can take statements and manipulate them in certain ways that are guaranteed not to change whether the statements are true or false. Intermediate algebra introduces kids to the idea that such transformations are possible, that abstract rules for the can (and must!) be learned and applied with precision. If offers them an opportunity to learn to communicate that transformation with concision and clarity as well. All of these are immensely valuable parts of general education, doubly so if we think of education as the process of elevating the soul to contemplate truth and not just get a job.

Would replacing that instruction with a survey of statistics have the same implicit effects? I suspect not. If it would have the same implicit effects, we’d expect that students who have been unable to master the transformational rules of intermediate algebra would be unable to master the transformational rules of statistics. We don’t see that, so by modus tollens, perhaps what’s going on in statistics doesn’t exercise the same abstract reasoning muscles.

If we wanted to replace intermediate algebra, I suspect we’d be best off replacing it with a deep dive into formal logic. A semester of the categorical syllogism backed up by a semester of propositional logic (and for the precocious kids, digital logic). That gets you a complex, abstract, rigorous set of truth-preserving transformations that students can master. Unlike intermediate algebra, it doesn’t rely on successful mastery of the rules of arithmetic, so some kids don’t start behind the eight ball.

Dan Meyer

William, I like the critical approach to change you describe in the first paragraph.

However:

I’d propose that what unifies intermediate algebra is the idea of a truth-preserving transformation.

IA offers students lots of practice applying transformations to lots of different kinds of statements of equality. But those ideas are introduced in beginning algebra and further developed with planar transformations in geometry.

In fact, I’m struggling to think of a new idea the course introduces about truth-preserving transformation. It just asks, “Hey, how about you try those ideas out on polynomials with degree five? How about with logarithms?” Those are valuable experiences, but only for students who elect into them IMO.

William Carey

You won’t fine me carrying much water for spending much time on quartics or quintics until Calculus. In my Algebra II class we use “Algebra for the Practical Man” by Thompson as our text, who walks the kids through the derivation of the quadratic and cubic formulas, and then essentially ignores quartics and quintics beyond a warning that there be dragons.

> How about with logarithms?

I will schlep a lot of water for logarithms. It provides exactly the sort of 1% busting epiphanic moment you want if you teach them well. Here’s my progression for them:

Start with the concrete. Give the kids a worksheet that asks them to computer \log_{10} of a bunch of carefully chosen numbers using a calculator. Don’t explain anything about what \log_{10} means. (I use 1, 2, 3, 4, 6, 9, 10, 12, 16, 20, 30, 40, 100, 1000, 4000, and 10,000.

Pattern time. Ask them to look for structure and patters. Usually about ten talking in small groups, then brainstorming as a class. With 100% success, the class notices that there’s some sort of additive structure to logarithms. I say nothing.

Prediction time. What’s \log_{10} of 24? Of 60? Of 200? Of 240? Eventually the students realize that they can predict the values of logarithms using the additive structure.

Generalization time. Ask the kids about the logarithm of a times b. Again, the class always works out the theorem. A decent fraction of kids have an epiphanic moment here, the sort of thing that you want to erode the 1% business.

Why is that so valuable that everyone should do it? A couple of reasons:

1. The heuristic of moving from concrete to patterns to prediction to generalization is an enormously valuable one to internalize for everyone. It’s truly general education.

2. Logarithms in particular illustrate a nice move: identifying a truth preservation that operates differently after some sort of transformation. I don’t know that you get that in the lower algebraic forms. This pays off crazy when I teach stoichiometry and can explain the mole/gram equality relationships as analogously similar to logarithms.

3. Logarithms offer a lovely opportunity to talk about the relationships between truth-preserving rule sets. At first they seem arbitrary, but they flow necessarily from the properties of exponents, which themselves flow from the properties of multiplication, which themselves flow from the properties of addition. Exposing students to a concrete example of how complex rulesets can be related to one another and have important implications for one another is general education.

4. Even if logarithms don’t necessarily introduce new ideas about truth-preserving transformations (I think they do), practicing the internalization and application of those transformations is also general education, especially as they become more complex. How much of our adult life is soaking up some new rule set, figuring out how to navigate with it, and then undertaking projects that rely on it? Almost all of it.

There’s a guy whose work I’d love for you to read some time – Ravi Jain. There are scattered fragments available online, but when his book is out, I’d love to hear what you think of it. He presents well what I present haltingly and inarticulately.

J. Kim

Thank you for the post. I agree with you that there is a need to change the definition of mathematics that people experience because obviously, the majority of people feel discouraged by the math taught at schools. Only a few people in the society consider themselves as “math people,” and others perceive them as privileged. If the best use of algebra in adulthood is to teach kids algebra, why does the curriculum emphasize algebra so much? I personally like algebra and I want to teach algebra because I want to spread the fun in it, but I’m sure not everyone enjoys doing math. My plan for now is to incorporate as many engaging activities that model the real world into my lessons to trigger the interest in students. I will have to think what my role will be as a teacher to change the definition of math that my students will experience.

I agree with the problem you raise, but not with the solution you propose.

Algebra 2 is often a terrible course. It is consistently at the heart of anti-math diatribes, and a painful memory for many students. I will not defend such a course. But it doesn’t have to be that way. I wrote a defense of the course and a summary of how to improve it here:https://blog.mathed.page/2016/05/in-defense-of-algebra-2.html

I taught this better Algebra 2 for years. After taking that course, maybe half of our students were able to take Calculus with a reasonable rate of success. But I’m guessing only maybe 10% of them could solve the equations you list from the entrance exam you quote.

The emphasis on highly technical and supremely boring manipulations is terribly counterproductive, for all the reasons you list. But a humane Algebra 2, which is interesting, and keeps students’ STEM options open, is entirely possible. I know, because I taught such a course.

As for statistics as a replacement… I have an MA in math, and I’ll be honest: I don’t understand the underlying math of, say, standard deviation, or confidence interval formulas. Those things are usually taught as black-box techniques, in a way that is 100% opposite to the sort of teaching for understanding we all strive for when teaching math.

What it boils down to is that we need better teaching of high school math, whether algebra or statistics or any of the other possibilities. NCTM’s recent document (_Catalyzing Change_) launches an in-depth conversation on how to do that. It will take a little more work to sort it all out than merely dropping the Algebra 2 requirement.